equal
deleted
inserted
replaced
119 and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H" |
119 and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H" |
120 and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H" |
120 and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H" |
121 unfolding H'_def sum_def lin_def by blast |
121 unfolding H'_def sum_def lin_def by blast |
122 |
122 |
123 have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0 |
123 have ya: "y1 + y2 = y \<and> a1 + a2 = a" using E HE _ y x0 |
124 proof (rule decomp_H') txt_raw \<open>\label{decomp-H-use}\<close> |
124 proof (rule decomp_H') text_raw \<open>\label{decomp-H-use}\<close> |
125 from HE y1 y2 show "y1 + y2 \<in> H" |
125 from HE y1 y2 show "y1 + y2 \<in> H" |
126 by (rule subspace.add_closed) |
126 by (rule subspace.add_closed) |
127 from x0 and HE y y1 y2 |
127 from x0 and HE y y1 y2 |
128 have "x0 \<in> E" "y \<in> E" "y1 \<in> E" "y2 \<in> E" by auto |
128 have "x0 \<in> E" "y \<in> E" "y1 \<in> E" "y2 \<in> E" by auto |
129 with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2" |
129 with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2" |